Method and System for Identification and Mitigation of Errors in Non-Line-of-Sight Distance Estimation

ABSTRACT

Ultra-wide bandwidth (UWB) transmission is a promising technology for indoor localization due to its fine delay resolution and obstacle-penetration capabilities. However, the presence of walls and other obstacles present a significant challenge in terms of localization, as they result in positively biased distance estimates. Measurement campaigns with FCC-compliant UWB radios can quantify effects of non-line-of-sight (NLOS) propagation. Features of waveforms measured during a campaign can be extracted for use in distinguishing between NLOS and line-of-sight situations in embodiments of the present invention. Embodiments further include classification and regression methods based on machine learning that improve the localization performance while relying solely on the received signal. Applications for systems employing an example embodiment of the invention include indoor or outdoor search and recovery with high accuracy and low cost.

RELATED APPLICATIONS

This application is a divisional of U.S. application Ser. No.12/383,939, filed Mar. 30, 2009. The entire teachings of the aboveapplication(s) are incorporated herein by reference.

GOVERNMENT SUPPORT

The invention was supported, in whole or in part, by grants ANI-0335256and ECCS-0636519 from the National Science Foundation and YoungInvestigator Award N00014-1-0489 from the Office of Naval Research. TheGovernment has certain rights in the invention.

BACKGROUND OF THE INVENTION

Location-awareness is fast becoming a fundamental aspect of wirelessnetworks and will enable a myriad of applications, in both thecommercial and the military sector. Localization in harsh environmentsand accuracy-critical applications requires robust signaling,through-wall propagation, and high-resolution ranging capabilities, suchas ultra-wide bandwidth (UWB) transmission, which operates in the 3 to10 GHz band. In practical scenarios, however, a number of challengesremain before UWB localization and communication can be deployed.

SUMMARY OF THE INVENTION

Embodiments of the present invention include methods of and apparatusesfor characterizing a medium comprising learning a function that isnonparametric with respect to features representing aspects of awaveform transmitted between two wireless devices. In alternativeembodiments, the function may be feature-based, i.e., based on featuresof transmitted waveforms. The medium is characterized by evaluating thefunction with a feature vector extracted from a waveform received at afirst wireless device from a second wireless device. Function evaluationyields information about the waveform, including, but not limited to: anestimate of the distance between the first and second wireless devices;a decision regarding a presence of an obstruction between the first andsecond wireless devices; a likelihood of an obstruction between thefirst and second wireless devices; an estimate of distance estimationerror; and a distribution of distance estimation error.

This information may be reported to a user or a module configured to usethe reported information. For example, the module may be configured tomitigate distance estimate error by subtracting the distance estimateerror from the distance estimate.

In further embodiments, the function is learned using machine learningtechniques, such as support vector machine techniques or Gaussianprocesses techniques. Functions may be chosen from a classes offunctions including, but not limited to, polynomials and linearcombinations of kernel functions. Learning may include accessing data ina training database, including line-of-sight/non-line-of-sight labelsand feature vectors representing aspects of a waveform transmittedbetween at least one pair of wireless devices. The training database maybe assembled from waveforms transmitted between pairs of wirelessdevices distributed throughout a model environment.

Embodiments may employ first and second wireless devices that arematched pairs of wireless devices with compatible transceivers. Featuresrepresenting aspects of a waveform include, but are not limited to:estimated distance; energy of a received signal; maximum amplitudes;rise time; mean excess delay; root-mean-square delay spread; andkurtosis. In some embodiments, all the features may be extracted fromthe received waveform; in other embodiments, features may includeinformation from other sources, such a priori information about theenvironment or the transmitted waveform.

Further embodiments include methods of and apparatuses for supportingdistance estimation between wireless devices. Example methods includeevaluating an identification function to determine a presence orlikelihood of an obstruction between wireless devices. A mitigationfunction can be evaluated based on a result of evaluating theidentification function. Results of evaluating the mitigation functionmay then be used to provide an estimate of distance estimation error (ordistribution of distance estimation error) of an estimate of a distancebetween wireless devices to support distance estimation between wirelessdevices.

The identification and mitigation functions may learned using machinelearning techniques. In certain instances, the identification functionmay be learned using an identification training database that includesat least line-of-sight/non-line-of-sight labels of waveforms received ina training environment. The mitigation function may be learned using amitigation training database that includes at least true distances ofwaveforms received in a training environment. The identification andmitigation training databases may be coupled to identification andmitigation units, respectively; they may, in fact, be the same database.

For example, evaluating the identification function may includeproviding labels identifying a received waveform as being line-of-sight(LOS) or non-line-of-sight (NLOS); these labels may be used to determinewhether to evaluate the mitigation function. Still further embodimentsmay correct the estimate of the distance between wireless devices basedon results of evaluating the mitigation function. Similarly, theestimate of distance estimation error (or distribution of distanceestimation error) may be used to localize a wireless device.

In example embodiments, evaluating the identification and mitigationfunctions occurs in one of plural layers in a communications networkbetween wireless devices. The estimate or distribution of distanceestimation error may be forwarded to a different layer in thecommunications network.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing will be apparent from the following more particulardescription of example embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingembodiments of the present invention.

FIG. 1 is a schematic diagram illustrating an example wireless networkthat provides identification and mitigation for errors innon-line-of-sight (NLOS) distance estimation in an indoor officeenvironment.

FIG. 2A is a flow diagram illustrating learning and evaluating anonparametric function to report decisions relating to a presence ofobstructions between wireless devices according to embodiments of thepresent invention.

FIG. 2B is a flow diagram illustrating learning and evaluating anonparametric function to report decisions relating to a presence ofobstructions between wireless devices according to alternativeembodiments of the present invention.

FIG. 3A is a block diagram illustrating an obstruction determinationapparatus according to embodiments of the present invention.

FIG. 3B is a block diagram illustrating an alternative obstructiondetermination apparatus according to embodiments of the presentinvention.

FIG. 4A is a flow diagram illustrating learning and evaluating afeature-based function to report decisions relating to a presence ofobstructions between wireless devices according to embodiments of thepresent invention.

FIG. 4B is a flow diagram illustrating learning and evaluating afeature-based function to report decisions relating to a presence ofobstructions between wireless devices according to alternative exampleembodiments of the present invention.

FIG. 5A is a block diagram illustrating a distance estimation apparatusaccording to embodiments of the present invention.

FIG. 5B is a block diagram illustrating an alternative distanceestimation apparatus according to embodiments of the present invention.

FIG. 6A is a flow diagram illustrating evaluating identification andmitigation functions to report estimates of distance estimation errorsor distributions of distance estimation errors according to embodimentsof the present invention.

FIG. 6B is a flow diagram illustrating evaluating identification andmitigation functions to report estimates of distance estimation errorsor distributions of distance estimation errors according to alternativeembodiments of the present invention.

FIG. 7A is a block diagram illustrating a further alternative distanceestimation apparatus according to embodiments of the present invention.

FIG. 7B is a block diagram illustrating yet another alternative distanceestimation apparatus according to embodiments of the present invention.

FIG. 8 is schematic diagram illustrating clusters throughout an officeenvironment used for capturing training measurements between wirelessdevices under different signal propagation conditions.

FIG. 9 is a plot illustrating a cumulative distribution function (CDF)of ranging error for line-of-sight (LOS) and NLOS conditions.

FIG. 10 is a pair of plots illustrating amplitudes of received LOS(upper plot) and NLOS (lower plot) signals.

FIG. 11 is a CDF of ranging error for NLOS propagation condition before(dotted/dashed line) and after (dashed line) mitigation according toembodiments of the present invention.

FIG. 12 is a plot of outage probability for an example NLOSidentification and mitigation system with five anchors, where theprobability of a NLOS condition between anchors is 20%.

FIG. 13 is a plot of outage probability for an example NLOSidentification and mitigation system with five anchors, where theprobability of a NLOS condition between anchors is 80%.

FIG. 14 is a plot of outage probability for an example NLOSidentification and mitigation system with five anchors and a fixedallowable position error of e_(th)=2 m.

DETAILED DESCRIPTION OF THE INVENTION

A description of example embodiments of the invention follows.

The teachings of all patents, published applications and referencescited herein are incorporated by reference in their entirety.

Example embodiments of the present invention compensate fornon-line-of-sight (NLOS) propagation effects that corrupt measurementsmade using high-resolution localization systems. Typically, NLOSpropagation introduces positive biases in distance estimationcalculations, thus seriously affecting the localization performance.Typical harsh environments, such as enclosed areas, urban canyons, orunder trees canopies, inherently have a high occurrence of NLOSsituations. It is therefore useful to understand the impact of NLOSconditions on localization systems and to develop techniques thatcounter their effects.

A generic localization system comprises two stages: a ranging stage anda localization stage. In the ranging stage, signals are exchangedbetween UWB devices, based on which relative angles or distances can beestimated. Ranging is often based on estimating the time of arrival(TOA) or the round-trip time of arrival (RTOA) of a packet. Based on theTOA or the RTOA, the distance can be inferred when the signalpropagation speed is known. When the direct line-of-sight (LOS) pathbetween the devices is unobstructed, this velocity can be assumed to beconstant. On the other hand, under NLOS conditions, the direct signalpath is obstructed, so the signal arrives either by propagation througha material or via a reflected path. In both cases, the distance estimateis positively biased. In the former case, this bias is due to reducedpropagation speed, which depends on material permittivity andpermeability, both of which can be frequency-dependent. In the lattercase, the bias is due to the spatial configuration of reflectors.

Example embodiments of the present invention include identification andmitigation methods and systems to deal with ranging bias in NLOSconditions. For instance, disclosed identification methods include anon-parametric approach to non-line-of sight (NLOS) identification basedon machine learning. Because these embodiments are based on machinelearning, they (i) do not require a statistical characterization ofline-of-sight (LOS) and NLOS channels; (ii) are based on theultrawideband (UWB) channel impulse response (CIR), and thus avoid anylatency issues; and (iii) perform identification and mitigation under acommon framework. Further, the disclosed techniques are based on anextensive UWB measurement campaign performed in a typical officeenvironment with FCC-compliant UWB radios, not on statistical channelmodels, so the results shown in FIGS. 9-14 realistically indicatereal-world performance.

FIG. 1 is a diagram of a floor plan of an indoor office environment 100featuring various wireless devices that employ NLOS identification andmitigation. The wireless devices may be, for example, associated withindividual firemen during a search and rescue mission. Fixed wirelessdevices, or anchors 110 a-c, may have known positions. Mobile wirelessdevices, or agents 111 a and 111 b, typically have unknown positions.

In preferred embodiments, the anchors 110 a-c and agents 111 a, 111 bcan transmit and receive wideband signals, including UWB signals. Agiven pair of wireless devices may exchange wideband signals todetermine their separation distance by multiplying the measured theround-trip signal propagation time by the signal propagation speed, c.Agents 111 a and 111 b may use distance estimates to three or moreanchors 110 a-c to localize their positions within the network.

These signals may be LOS signals, such as an LOS signal 122, whichpropagates along an unobstructed path directly from the anchor 110 b tothe agent 111 a. Wireless devices may also transmit and receive NLOSsignals obstructed by walls 102 or other obstacles, such as anobstructed signal 124 transmitted from the anchor 110 a to the agent 111a. Because the obstructed signal 124 travels at lower speed through theobstacle than through air, the corresponding distance estimate is toobig. NLOS signals may also include signals that follow indirect pathsbetween wireless devices, such as reflected signal 126 transmitted fromthe anchor 110 c to the agent 111 a. For the reflected signal 126, thesignal propagation distance is greater than the straight-lineseparation, so the estimated distance is again too big.

Wireless devices 110 a-110 e of the present invention can identify NLOSsignals as such and mitigate the effects of obstacles, reflections,etc., on received signal. For example, identification and mitigation canbe used to more accurately estimate distances between pairs of wirelessdevices or to more accurately localize a wireless device using any threeother wireless devices.

FIG. 2A is a flow diagram illustrating learning (205) and evaluating(210) a nonparametric function 207 according to embodiments of thepresent invention. In preferred embodiments, the nonparametric function207 is learned (205) using machine learning techniques, such as supportvector machine (SVM) or Gaussian processes. Evaluating (210) thenonparametric function 207 yields a decision regarding or likelihood ofthe presence of an obstacle 212. This decision or likelihood 212 isreported (215), possibly to a user, another wireless device, or adifferent physical or logical layer in a wireless network includingmultiple wireless devices.

FIG. 2B is a flow diagram illustrating learning (260) and evaluating(275) a nonparametric function 262 according to alternative embodimentsof the present invention. In this example, a training database isassembled (255) from measurements of signals, or waveforms, transmittedbetween pairs of prototype wireless devices (or perhaps other devicesaltogether) at different locations throughout a training environment,such as an indoor office environment. The training database includesinformation about received waveforms including, but not limited to:estimated distance between wireless devices; received energy; maximumamplitude of the received waveform; waveform rise time; mean excessdelay; root-mean-square (RMS) delay spread; and kurtosis. The trainingdatabase may also include actual distance measurements and NLOS/LOScondition information.

Training data 257 is used to learn (260) a nonparametric function 262using machine learning techniques, such SVMs or Gaussian processes. Oncelearning (260) ends, the training database does not have to be accessedagain, although it may be used to further refine function performance orto learn a different function.

When learning (260) is complete, the nonparametric function 262 may beused to evaluate waveforms 267 received (265) at a wireless device.Feature vectors 272 are extracted (270) from the received waveform 267;example feature vectors 272 may include, but are not limited to:estimated distance between wireless devices; received energy; maximumamplitude of the received waveform; waveform rise time; mean excessdelay; RMS delay spread; and kurtosis.

The extracted feature vector 272 is used to evaluate (275) thenonparametric function 262, possibly according to the furtherdescription below. Evaluating (275) the nonparametric function 262 mayyield either a decision regarding the presence of an obstacle betweenwireless devices or a probability that such an obstacle exists 277. Thisdecision or likelihood 277 is reported (280), possibly to a user,another wireless device, or a different physical or logical layer in awireless network including multiple wireless devices.

In some embodiments, once the function is learned, no more trainingoccurs. Learning is done once, during development of the wirelessdevice, on a different computer than the wireless device itself.Function evaluation (275), however, is done by the wireless device everytime a waveform is received during normal operating conditions. To speedprocessing, the function may be evaluated (275) for new incoming signalsonly.

FIG. 3A is a block diagram illustrating an obstruction determinationdevice 300 according to embodiments of the present invention. Theobstruction determination device 300, which may operate according to theflow diagrams shown in FIGS. 2A and 2B, learns a nonparametric function312 with a learning unit 310 using machine learning techniques. Thelearning unit 310 forwards the learned nonparametric function 312 to anevaluating unit 315, which evaluates a feature vector extracted from areceived waveform 305 using the nonparametric function 312. Theevaluating unit 312 produces a decision 325 regarding the presence of anobstacle in the waveform's path (or possibly a probability of thepresence of an obstacle in the waveform's path) and forwards thedecision 325 (or probability) to a reporting unit 320. The reportingunit 320 forwards the decision to a user, processor, different wirelessdevice, or different layer in the wireless network through which thewaveform 305 was transmitted.

FIG. 3B is a block diagram illustrating an alternative obstructiondetermination apparatus 350 according to embodiments of the presentinvention. As shown in FIG. 3B, a learning unit 360 receives trainingdata, including LOS/NLOS labels 382 and feature vectors 384, from atraining database 380. Example feature vectors 384 may includeinformation collected from training signals, such as: estimated distancebetween wireless devices; received energy; maximum amplitude of thereceived waveform; waveform rise time; mean excess delay; RMS delayspread; and kurtosis.

The learning unit 360 uses the training data to learn a nonparametricfunction 362 using machine learning techniques, such as SVMs or Gaussianprocesses. The learning unit 360 forwards the learned nonparametricfunction 362 to an evaluating unit 365, which evaluates the function 362for a feature vector extracted from a waveform 355. By evaluating thefunction 362 for a given feature vector, the evaluating unit 360determines whether an obstacle obstructed the waveform's path. Theevaluating unit 362 forwards a decision or probability 375 regardingobstacles in the waveform's path to a reporting unit 370, which mayforward the decision or probability 375 to a user, different module,different layer, or memory.

FIG. 4A is a flow diagram illustrating learning (405) and evaluating(410) a feature-based function 407 according to embodiments of thepresent invention. In preferred embodiments, the feature-based function407 is learned (405) using machine learning techniques, such as supportvector machine (SVM) or Gaussian processes. Evaluating (410) thefeature-based function 407 yields an estimate of distance estimationerror or a distribution of a distance estimation error 412. The estimateof the distance estimation error 412 may be derived from thedistribution of the distance estimation error; for example, it may bethe mean or median of the distribution. This error or distribution 412is reported (415), possibly to a user, another wireless device, or adifferent physical or logical layer in a wireless network includingmultiple wireless devices.

FIG. 4B is a flow diagram illustrating learning (455) a feature-basedfunction 457 using machine learning techniques according to alternativeembodiments of the present invention. A wireless device or othersuitable receiver receives (460) a waveform from which a distanceestimate is prepared (470) and a feature vector is extracted (465). Theextracted feature vector may include, but is not limited to: estimateddistance between wireless devices; received energy; maximum amplitude ofthe received waveform; waveform rise time; mean excess delay; RMS delayspread; and kurtosis.

Evaluating (475) the feature vector using the feature-based function 457produces an estimate of distance estimation error or a distribution ofdistance estimation error 477, which can be used to mitigate (480) errorin the distance estimate. The mitigated distance estimate and theestimate of distance estimation error (and/or the distribution ofdistance estimation error) 477 are reported (485, 490) to a user,another wireless device or processor, or another layer in the wirelessnetwork.

FIG. 5A is a block diagram illustrating a distance estimation apparatus500 according to embodiments of the present invention. The distanceestimation apparatus 500, which may operate according to the flowdiagrams shown in FIGS. 4A and 4B, learns a feature-based function 512with a learning unit 510 using machine learning techniques. The learningunit 510 forwards the learned nonparametric function 512 to anevaluating unit 515, which evaluates a feature vector extracted from areceived waveform 505 using the feature-based function 512. Theevaluating unit 512 produces an estimate of distance estimation error(or a distance estimation error distribution) 525 and forwards the errorestimate (distribution) 525 to a reporting unit 520. The reporting unit520 forwards the error estimate (distribution) 525 to a user, processor,different wireless device, or different layer in the wireless networkthrough which the waveform 505 was transmitted.

FIG. 5B is a block diagram illustrating an alternative distanceestimation apparatus 550 according to embodiments of the presentinvention. The distance estimation apparatus 550 receives a waveform555, then extracts a feature vector from the received waveform. Anevaluation unit 565 evaluates the feature vector with a feature-basedfunction 562, learned by a learning unit 560 using machine learningtechniques, such as SVMs and Gaussian processes. The evaluation unit 565forwards distance estimation error information 575, such as errorestimate or distribution, to a reporting unit 570, which reports theerror information 575, and a mitigation unit 580. The mitigation unit580 uses the error information 575 to mitigate error in estimateddistance; the mitigation unit 580 then reports the mitigated distanceestimate 585.

FIG. 6A is a flow diagram illustrating evaluating identification (605)and mitigation (610) functions according to embodiments of the presentinvention. The identification function is evaluated using features 603extracted from a received waveform. The features may be arranged in afeature vector and may include, but are not limited to: estimateddistance between wireless devices; received energy; maximum amplitude ofthe received waveform; waveform rise time; mean excess delay; RMS delayspread; and kurtosis.

Next, the mitigation function is evaluated (610) using results 612 fromevaluating (605) the identification function to produce, for example, anestimate of distance estimation error or a distribution of distanceestimation errors. The mitigation function results 612 are then provided(615) to a user, wireless device, localization system, or differentlayer in the wireless network.

FIG. 6B is a flow diagram illustrating an alternative method ofevaluating identification (655) and mitigation (665) functions, wherethe identification and mitigation functions are each functions ofwaveform features. The identification and mitigation functions may beseparate functions learned using machine learning techniques, such asthose described in greater detail below. The identification function maybe learned using a database that includes LOS and NLOS labels oftraining waveforms, as described above. The mitigation function may belearned using true distances of received training waveforms, possiblystored in the same database used to learn the identification function.Each function may be learned once before evaluation (655).

The identification function is evaluated (655) using features 653extracted from a received waveform to determine whether or not anobstruction exists between the pair of wireless devices associated withthe received waveform. If evaluating (655) the identification functionshows a strong likelihood that an obstruction exists, the waveform isidentified as an NLOS waveform (660). Otherwise, the waveform isidentified as an LOS waveform (660).

For NLOS waveforms, the mitigation function is evaluated (665) usingresults from evaluating (655) the identification function to produce,for example, an estimate of distance estimation error or a distributionof distance estimation errors. The mitigation function results 667 arethen used to correct (670) error in the distance estimate. For example,positively biased offset errors may be subtracted from the distanceestimate. In some embodiments, distance estimates for LOS waveforms maybe corrected (670) without being mitigated (665).

The corrected distances estimates for both LOS and NLOS waveforms maythen be used to localize (675) a wireless device within a wirelessnetwork. Localization (675) requires inputs from three or more devices(or three or more device locations) to triangulate the location of agiven wireless device relative to other devices on the network. Usingthe estimate of distance estimation error or distribution of distanceestimation error improves localization precision. Localization, errorinformation, and distance estimates are then reported to a differentlayer in the network.

FIG. 7A is a block diagram illustrating a distance estimation apparatus700 according to embodiments of the present invention. The distanceestimation apparatus 700, which may operate according to the flowdiagrams shown in FIGS. 6A and 6B, evaluates identification andmitigation functions learned using machine learning techniques. Anidentification unit 710 and a mitigation unit 715 each evaluate afeature vector extracted from a received waveform 705. The mitigationunit 715 may condition its evaluation on data from the identificationunit 710 to produce an estimate of distance estimation error (or adistance estimation error distribution) 725. The mitigation unit 715 andforwards the error estimate (distribution) 725 to a reporting unit 720,which forwards the error estimate (distribution) 725 to a user,processor, different wireless device, or different layer in the wirelessnetwork through which the waveform 705 was transmitted.

FIG. 7B is a block diagram illustrating another alternative distanceestimation apparatus 750, which may operate according to the flowdiagrams shown in FIGS. 6A and 6B. The distance estimation apparatus 750evaluates identification and mitigation functions learned using machinelearning techniques, such as SVMs and Gaussian processes. Theidentification and mitigation functions may be learned using LOS/NLOSlabels 762 and true distances 768, respectively, that are stored in anidentification database 763 and a mitigation database 767, respectively.An identification unit 760 evaluates a feature vector extracted from areceived waveform 755, then forwards an LOS/NLOS label 762 to amitigation unit 765.

If the label 762 identifies the waveform 755 as an NLOS signal, themitigation unit 765 mitigates NLOS effects, then forwards mitigateddistance data to a reporting unit 770, a correction unit 780, and alocalization unit 790. Otherwise, the mitigation unit 765 may forwardraw distance data directly to units 770, 780, and 790. The reportingunit 770 reports estimates of distance estimation error (or adistributions of distance estimation error) 775; the correction unit 780corrects error in the distance estimate to produce a corrected distanceestimate 785; and the localization unit 790 localizes a wireless deviceassociated with the waveform 755 using data from multiple devices ormultiple device locations to produce a localization estimate 795.

Problem Statement and System Model

This section includes a description of ranging and localization and thecorresponding need for NLOS identification and mitigation. A networkconsists of two types of nodes: anchors are nodes with known positions,while agents are nodes with unknown positions. For notationalconvenience, the focus here is from the point of view of a single agent,with unknown position p, surrounded by N anchors, with positions, p_(i),i=1, . . . , N. The distance between the agent and anchor i is denotedby d_(i)=|p−p_(i)| and the agent's estimate of this distance by{circumflex over (d)}_(i). The ranging error is ε_(i)={circumflex over(d)}_(i)−d_(i), and its estimate is {circumflex over (ε)}_(i). Thechannel condition between the agent and anchor i is λ_(i)ε{LOS, NLOS},and its estimate is {circumflex over (λ)}_(i). The mitigated distanceestimate of d_(i) is {circumflex over (d)}_(i) ^(m)={circumflex over(d)}_(i)−{circumflex over (ε)}_(i). The ranging error after mitigationis defined as ε_(i) ^(m)={circumflex over (d)}_(i) ^(m)−d_(i).

As shown below, localization requires knowledge of distance estimatesand anchor positions. For that reason, a set of useful neighbors Γconsisting of couples (p_(i), {circumflex over (d)}_(i)) can be useful.

Ranging Procedure

Suitable UWB radios include those that use a round-trip time of arrival(RTOA) method to perform ranging between two nodes without a common timereference: the first node, the requester, sends a ranging request to thesecond node, the responder. The ranging request consists of a packetwith a time stamp t_(Req,send) corresponding to the instant the requestis sent. The value of t_(Req,send) is expressed in the time reference ofthe requester. The responder receives the packet at time t_(Req,rec) andsends a second packet back to the first node at timet_(Res,send)=t_(Res,rev)+Δ. This packet is called a ranging response andcontains information about the duration of the interval Δ. The requesterreceives the ranging response at time t_(Req,rec) and is able toestimate the round-trip time using only its own internal clockreference:

$\begin{matrix}{{\hat{d}}_{{Res}->{Req}} = {\frac{1}{2} \times \upsilon \times ( {t_{{Req},{rec}} - t_{{Req},{send}} - \Delta} )}} & (1)\end{matrix}$

where v is the signal propagation speed. Typically, v is taken to equalc, the speed of light, but this becomes inaccurate under NLOSpropagation.

Localization Procedure

Once the agent has estimated distances with respect to N≧3 anchor nodes,it can estimate its position using Γ={p_(i),{circumflex over(d)}_(i)|1≦i≦N}. Many methods can achieve this goal, including the leastsquares (LS) technique, which is simple and requires no assumptionsregarding ranging errors. The agent can infer its position by solvingthe LS cost function,

$\begin{matrix}{\hat{p} = {\arg \; \underset{P}{\; \min}{\sum\limits_{i = 1}^{N}\; {( {{\hat{d}}_{i} - {{\hat{p} - p_{i}}}} )^{2}.}}}} & (2)\end{matrix}$

This optimization problem can be solved numerically using steepestdescent. In particular, setting the derivative of the cost function withrespect to p to zero leads to the following iterative procedure:

$\begin{matrix}{{\hat{p}}^{(l)} = {{\hat{p}}^{({l - 1})} + {\delta {\sum\limits_{i = 1}^{N}{( {{\hat{d}}_{i} - {{{\hat{p}}^{({l - 1})} - p_{i}}}} )e_{i}^{({l - 1})}}}}}} & (3)\end{matrix}$

where e_(i) ^((l-1)) is a unit vector oriented from p_(i) to {circumflexover (p)}_(i) ^((l-1)) and δ is a step size controlling the convergencespeed. Equation (3) can be interpreted as follows: every term in thesummation is zero when the distance estimate {circumflex over (d)}_(i)matches the distance between the estimates ∥{circumflex over (p)}_(i)^((l-1))−{circumflex over (p)}_(i)∥. When the distance between theestimates is larger than the distance estimate, the LS procedurecorrects this by moving the estimated position of the agent towardsanchor i. Conversely, when the distance between the estimates is smallerthan the distance estimate {circumflex over (d)}_(i), the LS procedurecorrects this by moving the estimated position of the agent away fromanchor i.

Sources of Error

Localization leads to erroneous results when the ranging errors arelarge. In practice the estimated distances are not equal to the truedistances, because of a number of effects, including thermal noise,multipath propagation, obstructions, and ranging artifacts.Additionally, the direct path between requester and responder may beobstructed, leading to NLOS signals. In NLOS conditions, the direct pathis either attenuated, due to through-material propagation, or completelyblocked. In the former case, the distance estimates are positivelybiased due to the reduced propagation speed (i.e., less than theexpected c). In the latter case, the distance estimate is alsopositively biased, as it corresponds to the first reflected path. Thesebias effects can be accounted for in either the ranging phase or thelocalization phase.

Embodiments of the present invention include techniques that work duringthe ranging phase to identify and mitigate the effects of NLOS signals.In NLOS identification, the terms in Eq. (3) corresponding to NLOSdistance estimates are omitted. In NLOS mitigation, the distanceestimate corresponding to NLOS signals are corrected, resulting in arange estimate that is more accurate. The localization equation, Eq.(3), can then adopt different strategies, depending on the quality andthe quantity of available range estimates.

Measurement Campaigns and Training Databases

FIG. 8 shows topological organization of clusters within an indooroffice environment at the Massachusetts Institute of Technology thatwere used by the Wireless Communications Research Group for a UWBLOS/NLOS measurement campaign. As shown in FIG. 8, several offices,hallways, one laboratory, and a large lobby constituted the physicalsetting of this campaign. The scope of the campaign matched theso-called indoor office environment. The Wireless CommunicationsResearch Group conducted the measurement campaign by collectingwaveforms sent from a wireless transmitter to a wireless receiver at avariety of positions and propagation conditions in the indoor officeenvironment. The associated range estimate and the actual distance werealso recorded for each waveform. The waveforms were then post-processedin order to reduce dependencies on the specific hardware.

The measurements were made using two Time Domain Corporation PulsOn 210(P210) radios capable of performing communications and ranging using UWBsignals. These radios are off-the-shelf transceivers that comply withthe emission limit set forth by the FCC. Specifically, the 10 dBbandwidth spans the band from 3.1 GHz to 6.3 GHz. The P210 radio isequipped with a bottom-fed planar elliptical antenna known as theBroadSpec P200. This kind of dipole is well-matched and radiationefficient. Most importantly, the antenna is omni-directional and thussuited for ad-hoc networks with arbitrary azimuthal orientation. Theradio runs a RTOA ranging protocol and is capable of capturing awaveform while performing the ranging procedure.

Measurements were take at over one hundred points in the considered areausing radios, or nodes, mounted on top of plastic carts at a height of90 cm above the ground. Points were placed randomly, but were restrictedto areas which are accessible by the carts. The distance between thetransmitter and receiver varied from roughly 0.6 m up to 18 m. In abouthalf the positions, obstacles between the transmitter and receivercreated NLOS conditions. The measurement points were grouped intoclusters, as shown in FIG. 8, where a given cluster corresponds to aroom or a region of a hallway. Each point belongs only to a singlecluster. Overall, around one thousand point-to-point measurements wereperformed, including measurements within a single cluster and betweenneighboring clusters. For each pair of points, several receivedwaveforms and distance estimates were recorded, along with the actualdistance. During each measurement, the radios remained stationary andcare was taken to limit movement of other objects in the nearbysurroundings.

A database was created from measurements collected during themeasurement campaign and used to develop and evaluate the proposedidentification and mitigation techniques. The database included 1024measurements: 512 LOS and 512 NLOS. The term LOS is used to denote theexistence of a visual LOS. Specifically, a measurement is labeled as LOSwhen the straight line between the transmitting and receiving antenna isunobstructed. Each waveform r(t) is affected by thermal noise andsampled at T_(sample)=41.3 ps over an observation window of 190 ns. Theranging estimate was obtained by a RTOA process created by Time DomainCorp. and embedded on the radio. The actual position of the radio duringeach measurement was manually recorded, and the ranging error wascalculated with the aid of a computer-aided design (CAD) software. Thecollected waveforms were then aligned in the delay domain using a simplethreshold-based method for leading edge detection. The alignment processcreates a common time reference which enables the subsequent extractionof the metrics used for identification and mitigation.

NLOS Identification and Mitigation

FIG. 9 shows the empirical cumulative distribution functions (CDFs) ofthe ranging error under the two different channel conditions. Thecollected measurement data shows that NLOS propagation conditionssignificantly impact ranging performance. In LOS conditions, the rangingerror is less than a meter more than 95% of the time. In the NLOS case,on the other hand, the ranging error is less than a meter less than 30%of the time. Clearly, LOS and NLOS range estimates have very differentcharacteristics.

FIGS. 10-14 show result of mitigation using techniques developed belowto distinguish between LOS and NLOS situations and to mitigate thepositive biases present in NLOS ranges estimates. The techniques includenon-parametric techniques, and rely on least-squares support-vectormachines (LS-SVM). The derivations below include a description offeatures used to distinguish between LOS and NLOS situations. This isfollowed by a brief introduction into LS-SVM and a description of howLS-SVM can be used for NLOS identification and mitigation in alocalization application.

FIG. 10 shows signals received under LOS (upper plot) and NLOS (lowerplot) conditions. Although there is wide variation in signalcharacteristics, a number of features can be extracted from everyreceived waveform r(t); these features (collectively, “a featurevector”) capture the salient differences. In general, NLOS signals areconsiderably more attenuated and present smaller energy and amplitudedue to reflections or obstructions. In the LOS case, the strongest pathcorresponds to the first path and the received signal presents a steeprise. In the NLOS case, some weak components precede the strongest path,resulting in a longer rise time. The RMS delay spread, which capturesthe temporal dispersion of the energy of the signal due to the multipathchannel, is larger in NLOS signals.

Taking these considerations into account, the extracted features mayinclude:

1. Energy of the received signal:

$\begin{matrix}{ɛ_{r} = {\int_{- \infty}^{+ \infty}{{{r(t)}}^{2}{t}}}} & (4)\end{matrix}$

2. Maximum amplitude of the received signal:

$\begin{matrix}{r_{\max} = {\max\limits_{t}{{r(t)}}}} & (5)\end{matrix}$

3. Rise time:

t _(rise) =t _(H) −t _(L)  (6)

where

t_(L)=min{t:|r(t)|≧ασ_(n)}

t_(H)=min{t:|r(t)|≧βr_(max)},

where σ_(n) is the standard deviation of the thermal noise. The valuesof α>0 and 0<β≦1 are chosen empirically; here, α=6 and β=0.6.

4. Mean excess delay:

$\begin{matrix}{\tau_{MED} = {\int_{- \infty}^{+ \infty}{t\; {\psi (t)}{t}}}} & (7)\end{matrix}$

where ψ(t)=|r(t)|²/E_(r).

5. RMS delay spread:

$\begin{matrix}{\tau_{RMS} = {\int_{- \infty}^{+ \infty}{( {t - \tau_{m}} )^{2}{\psi (t)}{t}}}} & (8)\end{matrix}$

6. Kurtosis:

$\begin{matrix}{\kappa = \frac{\frac{1}{T}{\int_{T}^{\;}{( {{{r(t)}} - \mu_{r}} )^{4}{t}}}}{\sigma_{r}^{4}}} & (9)\end{matrix}$

where μ_(|r|)=(1/T)∫_(T)|r(t)|dt and σ_(|r|)=(1/T)∫_(T)(|r(t)|−μ_(|r|))²dt.

These features can be learned and evaluated with machine learningtechniques, including support vector machines (SVMs), a supervisedlearning technique used both for classification and regression problems.SVMs are robust, have a rigorous underpinning, require few user-definedparameters, and have superior performance compared to other techniques,such as neural networks. LS-SVM is a low-complexity variation of thestandard SVM that has been applied successfully to classification andregression problems.

Classification:

A linear classifier is a function □^(n)→{−1,+1} of the form

l(x)=sign[y(x)]  (10)

with

y(x)=w ^(T)φ(x)+b  (11)

where φ(•) is a predetermined function, and w and b are unknownparameters of the classifier. These parameters are estimated based onthe training set {x_(k),l_(k)}_(k=1) ^(N), with inputs x_(k)ε□^(n) andlabels l_(k)ε{-1,+1}.

The SVM classifier is a maximum-margin classifier, obtained by solvingthe following optimization problem:

$\begin{matrix}{{\arg \mspace{11mu} {\min\limits_{w,b,\xi}{\frac{1}{2}{w}^{2}}}} + {\gamma {\sum\limits_{k = 1}^{N}\xi_{k}}}} & (12)\end{matrix}$s.t. l _(k) y(x)≧1−ξ_(k) ,∀k,  (13)

ξ_(k)≧0,∀k,  (14)

where y controls the trade-off between minimizing training errors andmodel complexity. (The margin is given by 1/∥w∥, and is defined as thesmallest distance between the decision boundary w^(T)φ(x)+b=0 and any ofthe training samples φ(x_(k)).) The Lagrangian dual turns out to be aquadratic program (QP). The LS-SVM replaces the inequality in Eq. (13)by an equality:

$\begin{matrix}{{\arg \mspace{11mu} {\min\limits_{w,b,e}{\frac{1}{2}{w}^{2}}}} + {\gamma \frac{1}{2}{e}^{2}}} & (15) \\{{{{s.t.\mspace{14mu} l_{k}}{y(x)}} \geq {1 - e_{k}}},{\text{∀}{k.}}} & (16)\end{matrix}$

The Lagrangian dual is now a linear program (LP):

$\begin{matrix}{{\begin{bmatrix}0 & l^{T} \\l & {\Omega + {I/\gamma}}\end{bmatrix}\begin{bmatrix}b \\\alpha\end{bmatrix}} = \begin{bmatrix}0 \\1_{N}\end{bmatrix}} & (17)\end{matrix}$

where Ω is an N×N matrix with Ω_(kl)=y_(k)y_(l)K(x_(k), x_(l)) whereK(x_(k), x_(l))=φ(x_(k))^(T)φ(x_(l)) is the kernel function. Solving theLP given in Eq. (17) yields the LS-SVM classifier:

$\begin{matrix}{{l(x)} = {{{sign}\mspace{11mu}\lbrack {{\sum\limits_{k = 1}^{N}{\alpha_{k}l_{k}{K( {x,x_{k}} )}}} + b} \rbrack}.}} & (18)\end{matrix}$

Regression:

A linear regressor is a function □^(n)→□ of the form

y(x)=w ^(T)φ(x)+b  (19)

where φ(•) is a predetermined function, and w and b are unknownparameters of the regressor. These parameters are estimated based on thetraining set {x_(k), l_(k)}_(k=1) ^(N), with inputs x_(k)ε□^(n) andoutputs y_(k)ε□. The LS-SVM regressor is obtained by solving thefollowing optimization problem:

$\begin{matrix}{{\arg \mspace{11mu} {\min\limits_{w,b,e}{\frac{1}{2}{w}^{2}}}} + {\gamma \frac{1}{2}{e}^{2}}} & (20) \\{{{s.t.\mspace{14mu} y} = {{y(x)} + e_{k}}},{\text{∀}k},} & (21)\end{matrix}$

where y controls the trade-off between minimizing training errors andmodel complexity. The Lagrangian dual is again a LP:

$\begin{matrix}{{\begin{bmatrix}0 & 1^{T} \\1 & {\Omega + {I/\gamma}}\end{bmatrix}\begin{bmatrix}b \\\alpha\end{bmatrix}} = \begin{bmatrix}0 \\y\end{bmatrix}} & (22)\end{matrix}$

with corresponding LS-SVM regressor

$\begin{matrix}{{{y(x)} = {{\sum\limits_{k = 1}^{N}{\alpha_{k}{K( {x,x_{k}} )}}} + b}},} & (23)\end{matrix}$

where α and b are the solution of the linear system in Eq. (22).

Since it is not possible to determine parametric joint distributions ofthe features given above for LOS and NLOS conditions, embodiments of thepresent invention use non-parametric LS-SVM instead. A 10-foldcross-validation is used to assess the performance of the features andthe SVM. This allows measurement of the performance of LS-SVM forcertain features and for subsets of the available features.

Classification:

A LS-SVM classifier is trained to distinguish between LOS and NLOS withinputs x_(k) (for training examples k) and corresponding labels l_(k)=+1when λ_(k)=LOS and l_(k)=−1 when λ_(k)=NLOS. The input x_(k) is composedof a subset of the following features: energy of the received signal;the maximum amplitude of the received signal; rise time; mean excessdelay; RMS delay spread; and kurtosis. A trade-off between classifiercomplexity and performance can be made by using a different size featuresubset.

Regression:

A LS-SVM regressor is trained to mitigate the effect of NLOS propagationwith inputs x_(k) (for training examples k, which were classified asbeing NLOS), and corresponding outputs y_(k)=ε_(k). Analogous to theclassification case, x_(k) is composed of the range estimate,{circumflex over (d)}_(k), and a subset of the following features:energy of the received signal; the maximum amplitude of the receivedsignal; rise time; mean excess delay; RMS delay spread; and kurtosis.Again, the performance achieved by the regressor depends on the size ofthe feature subset and the combination of features used.

Localization Strategies

The LS-SVM classifier and regressor allow development of the followinglocalization strategies: i) localization via identification, where onlyclassification is employed; ii) localization via identification andmitigation, where the received waveform is first classified and errormitigation is performed only on NLOS estimates; and iii) a hybridapproach which discards mitigated NLOS estimates when a sufficientnumber of LOS estimates are present.

In the standard strategy, all the range estimates {circumflex over(d)}_(i) from neighboring anchor nodes are used by the LS procedure forlocalization. In other words,

Γ_(S)={(p _(i) ,{circumflex over (d)} _(i)):1≦i≦N}.  (24)

In the identification strategy, signals associated with range estimatesare classified as LOS/NLOS using the LS-SVM classifier. Range estimatesare used in localization only if the associated signal was classified asLOS, while NLOS signals are discarded:

Γ_(I)={(p _(i) ,{circumflex over (d)} _(i)):1≦i≦N,{circumflex over (λ)}_(i) =LOS}.  (25)

Whenever the cardinality of Γ_(I) is less than three, the agent isunable to localize. In this case, the localization error is set to +∞.Note that measurements from three different points (e.g., three anchornodes) are needed to localize an agent in two-dimensions.

The identification and mitigation strategy is an extension to theprevious strategy, where the received signal is first classified as LOSor NLOS, and then mitigation is applied to those signals where{circumflex over (λ)}_(i)=NLOS. For this case Γ_(IM)=Γ_(I)∪Γ_(M) where

Γ_(M)={(p _(i) ,{circumflex over (d)} _(i) ^(m)):1≦i≦N,{circumflex over(λ)} _(i)=NLOS}.  (26)

This approach is motivated by the observation that there is no need tomitigate LOS waveforms since their accuracy is already extremely high.

In the hybrid approach, range estimates are mitigated as in the previousstrategy. However, mitigated range estimates are only used when lessthan three LOS anchors are available:

$\begin{matrix}{\Gamma_{H} = \{ \begin{matrix}\Gamma_{I} & {{{if}\mspace{14mu} {\Gamma_{I}}} \geq 3} \\\Gamma_{IM} & {otherwise}\end{matrix} } & (27)\end{matrix}$

(In practice the angular separation of the anchors should besufficiently large to obtain an accurate estimate. If this is not thecase, more than three anchors may be needed.) This approach is motivatedby the fact that mitigated range estimates are often still less accuratethan LOS range estimates. Hence, only LOS range estimates should be usedwhen they allow an unambiguous location estimate to be made.

Performance Evaluation and Discussion

This section includes quantification of the performance of the LS-SVMclassifier and regressor and the four localization strategies givenabove. Identification is considered first, then mitigation, and finallylocalization. The relevant performance measures and quantitative detailsof how the results were obtained are provided for every technique.

TABLE I FALSE ALARM PROBABILITY (p_(F)), PROBABILITY OF MISSED DETECTION(PM) AND OVERALL ERROR PROBABILITY (P_(err)) FOR DIFFERENT NLOSIDENTIFICATION TECHNIQUES. THE SET S_(i) DENOTES THE SET OF i FEATURESWITH THE SMALLEST P_(err) Identification Technique p_(F) pM P_(err)Parametric method 0.184 0.143 0.164 LS-SVM using features 0.129 0.1520.141 LS-SVM S₁ = {r_(max)} 0.137 0.123 0.130 LS-SVM S₂ = {r_(max),t_(rise)} 0.092 0.109 0.100 LS-SVM S₃ = {ε_(r), t_(rise), κ} 0.082 0.0900.086 LS-SVM S₄ = {ε_(r), r_(max), t_(rise), κ} 0.082 0.090 0.086 LS-SVMS₅ = {ε_(r), r_(max), t_(rise), τ_(MED), κ} 0.086 0.090 0.088 LS-SVM S₆= {ε_(r), r_(max), t_(rise), τ_(MED), τ_(RMS), κ} 0.092 0.090 0.091

Table 1 shows false alarm probability (p_(F)) probability of misseddetection (p_(M)) and overall error probability (P_(err)) for differentNLOS identification techniques and feature set sizes. The set S_(i)denotes the set of i features with the smallest P_(err). The kernel usedhere is a radial basis function (RBF) kernel of the form K(x,x_(k))=exp(−∥x−x_(k)∥²). The training error/model complexity tradeoff isset to y=0.1. Features are first converted to the log domain to reducethe dynamic range.

For the sake of comparison, the performance of the parametricidentification technique from I. Guvenc et al., “NLOS identification andweighted least-squares localization for UWB systems using multipathchannel statistics,” EURASIP J. on Advances in Signal Processing, vol.2008, 2008. This reference relies on the mean excess delay, the RMSdelay spread, and the kurtosis of the waveform. For fair comparison,these features are extracted from the training database described above.The performance is measured in terms of the misclassification ratesP_(err)=(p_(F)+p_(M))/2, where p_(F) is the false alarm probability(i.e., deciding NLOS when the signal was LOS), and P_(M) is the missprobability (i.e., deciding LOS when the signal was NLOS).

LS-SVM significantly reduces the false alarm probability, compared tothe parametric approach. The best set of features reduces the missprobability and achieves a total correct classification rate of above90%, compared to roughly 85% of the time for the parametric approach.The feature set of size three provides the best performance. It turnsout that among all feature sets of size three (results not shown), sixsets yield roughly equal P_(err). All six of these sets contained risetime (t_(rise)), while four contained maximum amplitude (r_(max)),indicating that these two features play an important role. This is alsocorroborated by the presence of maximum amplitude and rise time in theselected sets listed in Table 1.

TABLE II MEAN OF RESIDUAL RANGING ERROR AND AVERAGE SQUARED RESIDUALRANGING ERROR FOR LS-SVR MITIGATION. THE SET S_(i) DENOTES THE SET OF iFEATURES WHICH ACHIEVES THE MINIMUM RMS RRE. Mitigation Technique withLS-SVR mean [m] RMS RRE [m] No Mitigation 2.6322 3.589 S₁ = {{circumflexover (d)}} −0.0004 1.718 S₂ = {κ, {circumflex over (d)}} −0.0042 1.572S₃ = {t_(rise), κ, {circumflex over (d)}} 0.0005 1.457 S₄ = {t_(rise),τ_(MED), κ, {circumflex over (d)}} 0.0029 1.433 S₅ = {ε_(r), t_(rise),τ_(MED), κ, {circumflex over (d)}} 0.0131 1.425 S₆ = {ε_(r), t_(rise),τ_(MED), τ_(RMS), κ, {circumflex over (d)}} 0.0181 1.419 S₇ = {ε_(r),r_(max), t_(rise), τ_(MED), τ_(RMS), κ, {circumflex over (d)}} 0.01801.425

Table 2 shows means of residual ranging error (RRE) and average squaredRRE for LS-SVM mitigation, including results for different feature setsizes. The performance is measured in terms of the RMS RRE,

$\sqrt{{1/N}{\sum\limits_{i = 1}^{N}( ɛ_{i}^{m} )^{2}}}.$

The set S_(i) denotes the set of i features which achieves the minimumRMS RRE. The kernel used here is K(x, x_(k))=exp(−∥x−x_(k)∥²/256). Thetraining error/model complexity tradeoff is set to y=10. Features arefirst converted to the log domain to reduce the dynamic range.

A detailed analysis of the experimental data indicates that large rangeestimates are likely to exhibit large positive ranging errors. Thismeans that {circumflex over (d)} is a useful feature by itself, asconfirmed by the table. Increasing the feature set size can furtherimprove the RMS RRE. The feature set of size six (energy; maximumamplitude; rise time; mean excess delay; RMS delay spread; and kurtosis)offers the best performance. FIGS. 11-14 show NLOS mitigation resultsachieved with feature set of size six.

FIG. 11 shows the CDF of the ranging error before and after mitigationusing this feature set. Without mitigation, around 30% of the NLOSwaveforms achieved an accuracy of less than one meter (|ε|<1). However,after the mitigation process, 60% of the cases have an accuracy lessthan 1 m.

Simulation Setup:

The localization performance for varying number of anchors (N) and avarying probability of NLOS condition 0≦P_(NLOS)≦1 is evaluated with anagent at a position p=(0, 0). For every anchor i (1≦i≦N), a waveform isdrawn from the training databases: waveforms are drawn from the NLOSdatabase with probability P_(NLOS) and drawn from the LOS database withprobability 1−P_(NLOS). The true distance d_(i) corresponding to thatwaveform are then used to place the ith anchor at position p_(i)={d_(i)sin [2π(i−1)/N], d_(i) cos [2π(i−1)/N]}, while the estimated distance{circumflex over (d)}_(i) is provided to the agent. The agent estimatesits position, based on a set of useful neighbors Γ, using the LSprocedure given above, with the median of the anchor positions as theinitial position estimate {circumflex over (p)}⁽⁰⁾.

Performance Measure:

The notion of outage probability is useful for capturing the accuracyand availability of localization. For certain scenarios (N and P_(NLOS))and certain allowable errors (say, e_(th)=2 m), the agent is said to bein outage when its position error ∥p−{circumflex over (p)}∥ exceedse_(th). The outage probability is then given by

P _(out)(e _(th))=

{

{∥p−{circumflex over (p)}∥≧e _(th)}}  (28)

where I{P} is the indicator function, which, for a proposition P, iszero when P is false and one otherwise. The expectation in Eq. (28) isevaluated through Monte Carlo simulations.

Results:

FIGS. 12 and 13 show outage probability as a function of the allowableerror e_(th) for the different localization methods using the simulationsetup described above evaluated with N=5 anchors for different values ofP_(NLOS). When P_(NLOS)=0.2 (FIG. 12), on average one anchor is NLOS,possibly deteriorating the performance. When identification is employed,only the reliable LOS anchors are used. The standard strategy suffersfrom performance degradation compared to the identification strategy.The identification strategy does lead to an error floor phenomenon, dueto the fact that the probability that less than three LOS anchors areavailable is approximately 0.06.

Identification with mitigation does not suffer from the error floor,since all the anchors are used, but leads to some performancedegradation for allowable errors below 1.5 m. The hybrid approachcombines the benefits of the previous two strategies and leads to thelowest outage probability at all allowable errors. When P_(NLOS)=0.8(FIG. 13), the identification strategy does not perform well, since theprobability that less than three LOS anchors are available isapproximately 0.94.

The standard strategy does not fare much better, and both areoutperformed by the identification and mitigation strategy. Again, thehybrid strategy leads to the best performance. As P_(NLOS)→1, the lattertwo strategies have very similar performance because most of the anchorstend to an NLOS condition, so that Γ_(H)≈Γ_(M).

FIG. 14 shows outage probability for a fixed value of error, e_(th)=2,as a function of P_(NLOS). The Identification strategy is useful onlywhen P_(NLOS) is very small, due to the error floor phenomenon.Identification with mitigation drastically improves the performance, andgives outages below 20%, even when all the anchors are in NLOS. Thehybrid approach can further improve the performance, especially when asignificant fraction of anchors are in LOS conditions.

It should be readily appreciated by those of ordinary skill in the artthat the aforementioned blocks are merely examples and that the presentinvention is in no way limited to the number of blocks or the orderingof blocks described above. For example, some of the illustrated flowdiagrams may be performed in an order other than that which isdescribed. It should be appreciated that not all of the illustrated flowdiagrams is required to be performed, that additional flow diagram(s)may be added, and that some may be substituted with other flowdiagram(s).

It should also be apparent that methods involved in the invention may beembodied in a computer program product that includes a computer readablemedium. For example, such a computer readable medium may be a read-onlymemory device, such as a CD-ROM disk or convention ROM devices, or arandom access memory, such as a hard drive device, computer diskette, ormemory having a computer readable program code stored thereon. Thecomputer may load the program code and execute it to perform some or allof the example operations described herein or equivalents thereof.

While this invention has been particularly shown and described withreferences to example embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the scope of the inventionencompassed by the appended claims.

What is claimed is:
 1. A method of estimating error in distancemeasurements, the method comprising: learning a function based onfeatures representing aspects of a waveform transmitted between trainingwireless devices and a true distance between the training wirelessdevices; estimating error in a distance measurement by evaluating thefunction for a feature vector extracted from a waveform received at afirst wireless device from a second wireless device to determine anestimate of distance estimation error or a distribution of distanceestimation error of a distance estimate between the first and secondwireless devices; and reporting the estimate of distance estimationerror or the distribution of the distance estimation error to a modulemaking use of the estimate of distance estimation error or thedistribution of the distance estimation error.
 2. The method as claimedin claim 1 further comprising mitigating error in the distance estimateby subtracting the estimate of the distance estimation error from thedistance estimate.
 3. The method as claimed in claim 1 wherein thefeatures representing aspects of a waveform or feature vector include,but are not limited to at least one of the following features: estimateddistance; energy of a received signal; maximum amplitude of the receivedsignal; rise time; mean excess delay; root-mean-square delay spread; andkurtosis.
 4. The method as claimed in claim 1 wherein learning afunction is performed using machine learning techniques including, butnot limited to: support vector machine techniques and Gaussian processestechniques.
 5. An apparatus for estimating error in distancemeasurements, the apparatus comprising: a learning unit configured tolearn a function based on features representing aspects of a waveformtransmitted between training wireless devices and a true distancebetween the training wireless devices; an evaluating unit configured toestimate error in distance measurements by evaluating the function for afeature vector extracted from a waveform received at a first wirelessdevice from a second wireless device to determine an estimate ofdistance estimation error or a distribution of distance estimation errorof a distance estimate between the first and second wireless devices;and a reporting unit configured to report the estimate of distanceestimation error or the distribution of the distance estimation error toa module making use of the distance estimation error or the distributionof the distance estimation error.
 6. The apparatus as claimed in claim 5further including a mitigation unit configured to mitigate error in thedistance estimate by subtracting the estimate of the distance estimateerror from the distance estimate.
 7. The apparatus as claimed in claim 5wherein the features representing aspects of a waveform or featurevector include, but are not limited to at least one of the followingfeatures: energy of a received signal; maximum amplitude of the receivedsignal; rise time; mean excess delay; root-mean-square delay spread; andkurtosis.
 8. The apparatus as claimed in claim 5 wherein the learningunit learns the function using machine learning techniques including,but not limited to: support vector machine techniques and Gaussianprocesses techniques.
 9. A method of supporting distance estimationbetween wireless devices, the method comprising: evaluating anidentification function to determine a presence or likelihood of anobstruction of a wireless waveform transmitted between wireless devices;evaluating a mitigation function based on a result of evaluating theidentification function; and based upon a result of evaluating themitigation function, providing an estimate of distance estimation erroror a distribution of distance estimation error of an estimate of adistance between wireless devices to support distance estimation betweenwireless devices.
 10. The method as claimed in claim 9 furthercomprising: learning the identification function usingline-of-sight/non-line-of-sight labels of waveforms received in atraining environment; and learning the mitigation function using truedistances of waveforms received in a training environment.
 11. Themethod as claimed in claim 9 further including: correcting the estimateof the distance between wireless devices based on a result of evaluatingthe mitigation function.
 12. The method as claimed in claim 9 furtherincluding: localizing at least one of the wireless devices using theestimate of distance estimation error or the distribution of distanceestimation error.
 13. The method as claimed in claim 9 whereinevaluating the identification and mitigation functions occurs in one ofplural layers in a communications network between wireless devices; andreporting the estimate of distance estimation error or the distributionof distance estimation error includes forwarding the distance estimationerror or the distribution of distance estimation error to a differentlayer in the communications network.
 14. The method as claimed in claim9 wherein evaluating the identification function includes providinglabels identifying a received waveform as being line-of-sight (LOS) ornon-line-of-sight (NLOS) and evaluating the mitigation function is basedon whether the label identifies the received waveform as being NLOS. 15.An apparatus of supporting distance estimation between wireless devices,the apparatus comprising: an identification unit configured to evaluatean identification function to determine a presence or likelihood of anobstruction of a wireless waveform transmitted between wireless devices;a mitigation unit configured to evaluate a mitigation function based ona result of evaluating the identification function; and a reporting unitconfigured to provide, based upon a result of evaluating the mitigationfunction, an estimate of distance estimation error or a distribution ofdistance estimation error of an estimate of a distance between wirelessdevices to support distance estimation between wireless devices.
 16. Theapparatus as claimed in claim 15 further comprising: an identificationtraining database coupled to the identification unit, the identificationtraining database including at least line-of-sight/non-line-of-sightlabels of waveforms received in a training environment; and a mitigationtraining database coupled to the mitigation unit, the mitigationtraining database including at least true distances of waveformsreceived in a training environment.
 17. The apparatus as claimed inclaim 15 further including: a correction unit configured to correct theestimate of the distance between wireless devices based on a result ofevaluating the mitigation function.
 18. The apparatus as claimed inclaim 15 further including: a localization unit configured to localizeat least one of the wireless devices using the estimate of distanceestimation error or the distribution of distance estimation error. 19.The apparatus as claimed in claim 15 wherein the identification andmitigation units operate in one of plural layers in a communicationsnetwork between wireless devices; and wherein the reporting unitforwards the estimate of the distance estimation error or thedistribution of distance estimation error to a different layer in thecommunications network.
 20. The apparatus as claimed in claim 15 whereinthe identification unit is further configured to provide a labelidentifying a received waveform as being line-of-sight (LOS) ornon-line-of-sight (NLOS) and the mitigation unit is further configuredto evaluate the mitigation function based on whether the labelidentifies the received waveform as being NLOS.